Quantum Principles and Mathematical Computability

Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then outline a quantum mechanical “algorithm” for one of the insoluble problems of mathematics, the Hilbert's tenth and equivalently the Turing halting problem. The key element of this algorithm is the computability and measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values.